Accurately Recover Global Quasiperiodic Systems by Finite Points

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-07-24 DOI:10.1137/23m1620247

Kai Jiang, Qi Zhou, Pingwen Zhang

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024.
Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.

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用有限点精确恢复全局准周期系统

SIAM 数值分析期刊》第 62 卷第 4 期第 1713-1735 页，2024 年 8 月。摘要。准周期系统与无理数有关，是没有衰减或平移不变性的空间填充结构。如何精确恢复这些系统，尤其是低规则性情况，是数值计算中的一大挑战。本文提出了一种新算法--有限点复原（FPR）方法，它既适用于连续情况，也适用于低规则情况，以解决这一难题。FPR 方法首先在准周期函数的低维定义域和高维环之间建立同构，然后在定义域中采用有限点插值技术恢复全局准周期系统，而无需提维。此外，我们还根据无理数的算术特性，开发了精确高效的有限点选择策略。我们提出了选择有限点的相应数学理论、收敛性分析和计算复杂性分析。数值实验证明了 FPR 方法在恢复连续准周期函数和片断常数斐波那契准晶方面的有效性和优越性，而现有的光谱方法在恢复片断常数准周期函数方面遇到了困难。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.